The Rule of 72 and Doubling Time

The drunken haze that I refer to as my ‘first year in University’ instilled within me three distinctly poignant lessons: that I did not want, in any way, shape, or form, to become a teacher; that one should never add water to a jar containing sodium just because there doesn’t seem to be enough fluid covering it; and that statistics are the bane of the layman.

One Sunday afternoon when my informal Dr. Who fan club couldn’t find a free venue in which to congregate, smoke, and watch our favorite show, I found myself sitting alone reading How to Lie with Statistics by Darrell Huff. It was a book that changed my life because, for the first time, I realized that mathematics could tell lies. Darrell Huff taught me that cold, hard, irrefutable mathematical reductions could invoke false beliefs, without need for any mathematical trickery whatsoever.

The problem is that math is amoral and unsocial; it just has no concept of right or wrong and it has absolutely no desire at all to communicate with us. If one wants to understand what math has to say then one must learn the language of math for, like all too many Anglophones, math simply refuses to communicate in the vernacular. You see, it isn’t so much that math tells lies but, rather, that it speaks its own language and cares not what meaning we glean from its pronouncements. The misrepresentations conveyed by math are actually the result of our own shortcomings. Consider the following two statements:

This portfolio will double every 10 years.

This portfolio will provide 7.2% annual growth.

Which one sounds like the better investment to you? If you are a layman of mathematics then you may be surprised to learn that both statements predict the same returns. This is an illustration of the rule of 72. The rule of 72 is a rule of thumb that aids in the prediction of doubling time. Essentially, if you divide 72 by the percentage of growth (72 divided by 7.2), the result will be the number of cycles (years) it takes for the original amount to double. If something increases by 7% per month, then it will double in roughly 10 months. If something increases by 10% per year, then it will take roughly 7.2 years to double.

This one little rule of thumb is critical for making decisions in today’s world, yet few people are even aware of it. When newspapers report that inflation for the year was 3.6%, most people just shrug that off as some discreet and irrelevant economic statistic. Inflation, however, represents the amount by which prices have gone up in the consumer market. Although 3.6% inflation may seem irrelevant, the realization that your grocery bill has doubled over the past 20 years can be startling.

When I was a wee child, a bottle of soda cost 25 cents; now the bottle is made of plastic and the price is a dollar. For those not inclined to do the math on this one, either inflation has been greater than 3.6% during my lifetime or it has been more than 40 years since my earliest memory of pulling a soda out of a vending machine. You see, at an annual inflation rate of 3.6%, it takes 20 years (72 divided by 3.6) for the price of soda to double. After 20 years, that 25 cent bottle came to cost 50 cents, and after another 20 years it came cost one dollar.

The very concept of doubling time itself is a difficult concept for most people to grasp. The most powerful illustration of doubling time that I’ve ever heard involves a tale about a man who invented the game of chess for his king. The king was so impressed that he offered the inventor anything that his heart desired. The inventor, being mischievous as well as mathematically inclined, told the king that he desired a precisely determined amount of rice. He stated that he wanted one grain of rice for the first square on the board, two for the next square, and then three, and so on, for every square on the board. The king laughed at the inventor’s seemingly paltry wish and granted that it should be so.

Now there are 64 squares on a chessboard, and after only 8 squares (1, 2, 4, 8, 16, 32, 64, 128 …) the king made a keen observation. The next number was going to be 256, and it so happens that 1+2+4+8+16+32+64+128=255. This, he accurately surmised, meant that each time the amount was doubled the result was one greater than the total number of grains that had already been granted. The inventor was kind enough to inform the king that this mathematical truth was going to play a big roll in the future of his kingdom.

By the end of the second row the court jester had taken several days to count the 32,768 grains of rice required for the 16th square. The king, becoming very concerned at the prospect of having to count so many grains of rice, asked the inventor if they could just start weighing the rice instead. The inventor agreed, and they determined that the 32,768 grains of rice weighed about 8 kilograms and so it was that they continued with the payments by weight. By the end of the 3rd row, the king was amazed to find that four carts had to be employed to hall in the 2,048 kilograms of rice required for the 24th square. By the end of the 4th row, however, things started looking grim for the king as cart after cart had to be driven into the castle for days in order to provide the necessary 524,288 kilograms of rice that were required for the 32nd square.

The king became furious as he pondered the toll impending on the second half of the chessboard, and he asked the inventor just how many grains of rice it would take to fulfill the entire contract. The inventor told the king that the number of grains of rice required for the 64th square would be 9,223,372,036,854,775,808. Since every time the amount was calculated for a square it would result in a number that was one greater than all the rice previously granted, the rice already granted by that time would be 9,223,372,036,854,775,807. The sum of the two numbers, he said, was 18,446,744,073,709,551,615 – or roughly 2 trillion metric tons of rice. The king seized upon the inventor and promptly cut off his head.

This little story about the rice and the chessboard has always stuck with me because of just how profoundly it illustrates the potential of exponential growth. Armed with this knowledge, and the rule of 72, one can actually begin to understand why so many economists and scientists are so preoccupied with studying trends. To this end, let us consider how short sighted the king’s murderous rage might have been.

As the king had accurately surmised, every time the number was doubled, the result was actually greater than the sum of all the previous numbers. The inventor had also been kind enough to inform the king that this mathematical truth was going to play a big roll in the future of his kingdom. The trouble, though, was that the king had no idea what the inventor had meant by that and the poor sod’s head was already rolling around the courtyard before the king had regained the presence of mind to ask for clarification.

The king called upon his barristers to find out if he was obligated by any other contracts that stipulated a doubling of payments and they assured him that he was not. The king then racked his brain to think of anything in his kingdom that had ever doubled. The only thing he could think of was the number of farms he had raided to keep his kingdom well fed. In the last year of his father’s reign, the old man had raided 40 farms. After the old man died, passing the crown to the foolish chessboard king, it took 7 long years to increase that number to 80, and he remembered it well for he had thrown a big celebration to commend his knights on having doubled their efforts. It suddenly dawned on him that it had been about 7 years since that glorious celebration, and this year’s plans were to raid 160 farms.

This didn’t seem quite the same as the problem with the chessboard, however, for the raids had never doubled in one year. The king consulted with the court scholar on the matter, and the scholar assured him that the number of raids only increased by a little over 10% each year so their was certainly no reason to worry about a repeat of the chessboard fiasco. (Reminder: by the rule of 72, a 10% annual growth rate produces a doubling time of roughly 7 years: 72 divided by 10.)

Unfortunately, although the scholar knew about the rule of 72, and had recently witnessed how doubling values lead to exponential growth, he just didn’t understand how such equations could impact something like farm raids. Since the king’s father had raided 40 farms the year before handing over the crown, the king himself, raiding 10% more each year, had kicked off his reign with 44 raids. In the 7th year of his reign he raided 80 farms, and in year 14 he planned to raid 160. How then, might the next 7 years look in terms of raids? These numbers are illustrated in the table below.

Farm Raids

Period 1

Period 2

Period 3

44

88

177

49

98

195

54

108

216

59

119

238

66

131

263

72

145

290

80

160

320

Totals:

424

849

1699

As you can see, not only does the number of annual farms raids move from 80 to 160, but the total number of farms raided in periods one and two also doubles. Furthermore, with the number of annual farm raids predicted to double again, to 320, the total number of raids for period 3 (1699) is actually greater than for the 14 years previous to that period (424+849=1273). Although the number of raids was only increasing by 10% per year, the number of farms that the king would need to raid in any given 7 year period would actually be greater than the sum total of all farms he had ever raided before that period, in the entire history of his kingdom. At some point, perhaps in his great-grandson’s reign, there simply wouldn’t be enough farms in all of India to raid. If you don’t believe this then please re-read the chessboard segment of this article.

In 1977, when U.S. President Jimmy Carter said that the world had used more oil in each of the preceding decades than had been consumed previous to those decades, in the history of the world, millions upon millions of people laughed openly. Hopefully the reader has gained enough from this article so as not to ever be moved to laughter by such ignorance, for even today, few people realize just how dramatically that consumption trend had to be altered. The fact is that we haven’t been able to double our oil usage in any 10 year period that followed that speech. Today we need to face the fact that we will never again double that number, for we’ve already consumed half of all that was ever available and we would need more than what is left to ever double our consumption again.

We must acknowledge, therefore, the importance of remaining diligent in observing trends in all forms of economic growth and consumption. Today, rather than raiding farms, we seek out new sources of energy as petroleum sources begin to decline. As we continue to double our energy consumption, however, we will find that there are only so many rivers to be dammed, only so much uranium to be mined, and only so much wind to be harvested. Even though the sun is, for us, an eternal source of energy, our ability to harvest that energy is dependant upon our ability to source out the materials required for that harvest. Eventually we must face the fact that there are a finite number of farms for us to raid.

Thanks for the link, Wassabi. I have actually seen that video and thought about embedding it at the bottom of my post, but I felt my article was already too long. Hopefully some others will click through for that lesson.

Would it be accurate to say math is amoral as opposed to immoral? I think math can and is used in a very immoral way on a daily basis. Any statistic can be manipulated to provide a desired result and I think that what we here about polls should be taken with a very wary grain of salt.